9 MEASURABLE FUNCTIONS 141 (13.9.15) If fis integrable and if A, B are two measurable sets such that A n B is negligible, then f fdp= {'fdfi + f/d/i. JAuB JA JB For <pAu B and <pA + <pB are then equivalent functions (12.7.3). Remark (13.9.16) If I is a compact interval [a, b"] in R, then for any regulated function / with compact support we have f f dX = f f(x) dx, where A is Jl Ja Lebesgue measure (13.8.8). If now/is an arbitrary regulated function defined on R, then / is measurable because it is the limit of the integrable functions f<Pin, where !„ = [ — «,«] (13.9.11). For/to be integrable, it is necessary and sufficient that lim f "\f(x)\ dx < + 00, and in that case we have n-xxjJ"" (fdA, = lim f+"f(x) dx ((13.8.2) and (13.8.4)); this integral is also denoted •> n-+ooj-" by f *f(x) dx. It can happen that the limit lim f *f(x) dx exists although J~n n-^J-n lim | °°|/(jc)| dx = + oo (this is the case, for example, if f(x) = (sin x)/x for n-*ooj~ °° x ^ 0). The interpretation of most of these "improper integrals" belongs to the theory of distributions (Section 17.5, Problem 4, and Chapter XXIII). For the reasons given above, if I c R is any interval with endpoints a, b (with a :g 6), we write | f(x) dx in place of | /WA, for any function / such that/cp, is integrable. The (positive) measure \JL is said to be bounded if the constant function 1 is integrable, or in other words (13.9.13), if ju*(X) = /**(!) infinite: the number XX) = n(l) = f d\i is then called the total mass of p (cf. (13,20)). jx (13.9.17) If \JL is bounded, then every bounded measurable function f: X ->> R is integrable (in particular, every bounded semicontinuous function is integrable) and (13.9.17.1) J/4, This follows directly from (13.9.13).A/(*) dfi(x);